This contribution deals with the creation of numerical models for the simulation of the dynamic characteristics of fractional-order control systems and their comparison with analytical models. We give the results of the comparison of dynamic properties in fractional- and integer-order systems with a controller, designed for an integer-order system as the best approximation to given fractional-order system. Other open questions are pointed out, which should be answered in this area of research.
This paper deals with fractional-order controllers. We outline mathematical description of fractional controllers and methods of their synthesis and application. Synthesis method is a modified root locus method for fractional-order systems and fractional-order controllers. In the next section we describe how to apply the fractional controller on control systems.
This paper is concerned with the concepts of regional controllability for the Riemann-Liouville time fractional diffusion systems of order $α\in(0,1)$. The characterizations of strategic actuators to achieve regional controllability are investigated when the control inputs emerge in the differential equations as distributed inputs. In the end, an approach to guarantee the regional controllability of the problems under consideration in the considered subregion with minimum energy control is described and successfully tested through two applications.
This paper deals with fractional-order controlled systems and fractional-order controllers in the discrete domain. The mathematical description by the fractional difference equations and properties of these systems are presented. A practical example for modelling the fractional-order control loop is shown and obtained results are discussed in conclusion.
In this paper, we study a class of fractional optimal control problems. A necessary condition for the existence of an optimal control is provided in the literature. It is commonly given as the existence of a solution of a fractional Pontryagin's system and the proof is based on the introduction of a Lagrange multiplier. Assuming an additional condition on these problems, we suggest a new presentation of this result with a proof using only classical mathematical tools adapted to the fractional case: calculus of variations, Gronwall's Lemma, Cauchy-Lipschitz Theorem and stability under perturbat…
This paper deals with fractional-order controlled systems and fractional-order controllers in the frequency domain. The mathematical description by fractional transfer functions and properties of these systems are presented. The new ways for modelling of fractional-order systems are illustrated with a numerical example and obtained results are discussed in conclusion.
We introduce a nonlocal control condition and the notion of approximate controllability for fractional order quasilinear control inclusions. Approximate controllability of a fractional control nonlocal delay quasilinear functional differential inclusion in a Hilbert space is studied. The results are obtained by using the fractional power of operators, multi-valued analysis, and Sadovskii's fixed point theorem. Main result gives an appropriate set of sufficient conditions for the considered system to be approximately controllable. As an example, a fractional partial nonlocal control functional …
This paper derives the Hamilton-Jacobi-Bellman equation of nonlinear optimal control problems for cost functions with fractional discount rate from the Bellman's principle of optimality. The fractional discount rate is described by Mittag-Leffler function that can be considered as a generalized exponential function.
This paper deals with the fractional-order controllers and their possible hardware realization based on PIC microprocessor and numerical algorithm coded in PIC Basic. The mathematical description of the digital fractional -order controllers and approximation in the discrete domain are presented. An example of realization of the particular case of digital fractional-order PID controller is shown and described.
A class of optimal control problems governed by linear fractional diffusion equation with control constraint is considered. We first establish some results on the existence of strong solution to the state equation and the existence of optimal solutions for the optimal control problem. Then we derive first-and second-order optimality conditions for locally optimal solutions to the problem.
Fractional control systems with the Caputo derivative are considered. The modified controllability Gramian and the minimum energy optimal control problem are investigated. Construction of minimizing steering controls for the modified energy functional are proposed.
We survey some representative results on fuzzy fractional differential equations, controllability, approximate controllability, optimal control, and optimal feedback control for several different kinds of fractional evolution equations. Optimality and relaxation of multiple control problems, described by nonlinear fractional differential equations with nonlocal control conditions in Banach spaces, are considered.
We study the existence and approximate controllability of a class of fractional nonlocal delay semilinear differential systems in a Hilbert space. The results are obtained by using semigroup theory, fractional calculus, and Schauder's fixed point theorem. Multi-delay controls and a fractional nonlocal condition are introduced. Furthermore, we present an appropriate set of sufficient conditions for the considered fractional nonlocal multi-delay control system to be approximately controllable. An example to illustrate the abstract results is given.
We review recent results obtained to solve fractional order optimal control problems with free terminal time and a dynamic constraint involving integer and fractional order derivatives. Some particular cases are studied in detail. A numerical scheme is given, based on expansion formulas for the fractional derivative. The efficiency of the method is illustrated through examples.
Problem of time-optimal control of linear systems with fractional dynamics is treated in the paper from the convex-analytic standpoint. A linear system of fractional differential equations involving Riemann--Liouville derivatives is considered. A method to construct a control function that brings trajectory of the system to the terminal state in the shortest time is proposed in terms of attainability sets and their support functions.
In this paper, we discuss a new general formulation of fractional optimal control problems whose performance index is in the fractional integral form and the dynamics are given by a set of fractional differential equations in the Caputo sense. We use a new approach to prove necessary conditions of optimality in the form of Pontryagin maximum principle for fractional nonlinear optimal control problems. Moreover, a new method based on a generalization of the Mittag-Leffler function is used to solving this class of fractional optimal control problems. A simple example is provided to illustrate th…
We study the regional controllability problem for delayed fractional control systems through the use of the standard Caputo derivative. First, we recall several fundamental results and introduce the family of fractional-order systems under consideration. Afterwards, we formulate the notion of regional controllability for fractional systems with control delays and give some of their important properties. Our main method consists in defining an attainable set, which allow us to prove exact and weak controllability. Moreover, main results include not only those of controllability but also a power…
In this paper, we consider the approximate controllability of partial differential equations with time derivatives of non-integer order via boundary control. We first show the unique existence of the solution under smooth boundary conditions using the eigenfunction expansion. Next we also study the dual system and apply their results to prove (i) the regularity of the solution with non-smooth boundary conditions and (ii) approximate controllability.
This paper is concerned with model reference adaptive controller design for a class of nonlinear fractional order systems. Recent works on this topic rarely include direct methods and they are mostly based on indirect methods where the frequency distributed model is used to prove the stability of the closed loop system. Since the chain rule cannot be applied in fractional derivations, in order to prove the lyapunov stability here fractional inequalities are used. Finally, by means of a numerical example, the controller performance is demonstrated.
In this paper, we are concerned with the regional boundary controllability of the Riemann-Liouville time fractional diffusion systems of order $α\in (0,1]$. The characterizations of strategic actuators are established when the systems studied are regionally boundary controllable. The determination of control to achieve regional boundary controllability with minimum energy is explored. We also show a connection between the regional internal controllability and regional boundary controllability. Several useful results for the optimal control from an implementation point of view are presented in …
We study an optimal control problem associated to a fractional nonlocal thermistor problem involving the ABC (Atangana-Baleanu-Caputo) fractional time derivative. We first prove the existence and uniqueness of solution. Then, we show that an optimal control exists. Moreover, we obtain the optimality system that characterizes the control.
The current article examines the approximate controllability problem for non-instantaneous impulsive fractional evolution equations of order $1<α<2$ with state-dependent delay in separable reflexive Banach spaces. In order to establish sufficient conditions for the approximate controllability of our problem, we first formulate the linear-regulator problem and obtain the optimal control in feedback form. By using this optimal control, we deduce the approximate controllability of the linear fractional control system of order $1<α<2$. Further, we derive sufficient conditions for the approximate c…
In this paper we propose a model predictive control scheme for constrained fractional-order discrete-time systems. We prove that all constraints are satisfied at all time instants and we prescribe conditions for the origin to be an asymptotically stable equilibrium point of the controlled system. We employ a finite-dimensional approximation of the original infinite-dimensional dynamics for which the approximation error can become arbitrarily small. We use the approximate dynamics to design a tube-based model predictive controller which steers the system state to a neighbourhood of the origin o…
This paper deals with designing a robust fixed-order dynamic output feedback controller for uncertain fractional order linear time invariant (FO-LTI) systems by means of linear matrix inequalities (LMIs). Our purpose is to design a low order controller that stabilizes the fractional-order linear system in the presence of model uncertainties. No limitative constraint on the state space matrices of the uncertain system is assumed in the design procedure. Furthermore, adopting the most complete model of linear controller, with direct feedthrough parameter, does not disturb the LMI-based approach …
We develop a simple and accurate method to solve fractional variational and fractional optimal control problems with dependence on Caputo and Riemann-Liouville operators. Using known formulas for computing fractional derivatives of polynomials, we rewrite the fractional functional dynamical optimization problem as a classical static optimization problem. The method for classical optimal control problems is called Ritz's method. Examples show that the proposed approach is more accurate than recent methods available in the literature.
We investigate Optimal Control Problems (OCP) for fractional systems involving fractional-time derivatives on time scales. The fractional-time derivatives and integrals are considered, on time scales, in the Riemann--Liouville sense. By using the Banach fixed point theorem, sufficient conditions for existence and uniqueness of solution to initial value problems described by fractional order differential equations on time scales are known. Here we consider a fractional OCP with a performance index given as a delta-integral function of both state and control variables, with time evolving on an a…
We study a stochastic control system involving both a standard and a fractional Brownian motion with Hurst parameter less than 1/2. We apply an anticipative Girsanov transformation to transform the system into another one, driven only by the standard Brownian motion with coefficients depending on both the fractional Brownian motion and the standard Brownian motion. We derive a maximum principle and the associated stochastic variational inequality, which both are generalizations of the classical case.
The present paper deals with the control problems governed by fractional non-instantaneous impulsive functional evolution equations with state-dependent delay involving Caputo fractional derivatives in Banach spaces. The main objective of this work is to formulate sufficient conditions for the approximate controllability of the considered system in separable reflexive Banach spaces. We have exploited the resolvent operator technique and Schauder's fixed point theorem in the proofs to achieve this goal. The approximate controllability of linear system is discussed in detail, which lacks in the …
We consider optimal control of fractional in time (subdiffusive, i.e., for $% 0<γ<1$) semilinear parabolic PDEs associated with various notions of diffusion operators in an unifying fashion. Under general assumptions on the nonlinearity we{~\textsf{first show}} the existence and regularity of solutions to the forward and the associated {\textsf{backward (adjoint)}} problems. In the second part, we prove existence of optimal {\textsf{controls% }} and characterize the associated {\textsf{first order}} optimality conditions. Several examples involving fractional in time (and some fractional in sp…
We present a method to solve fractional optimal control problems, where the dynamic depends on integer and Caputo fractional derivatives. Our approach consists to approximate the initial fractional order problem with a new one that involves integer derivatives only. The latter problem is then discretized, by application of finite differences, and solved numerically. We illustrate the effectiveness of the procedure with an example.
We design and analyze solution techniques for a linear-quadratic optimal control problem involving the integral fractional Laplacian. We derive existence and uniqueness results, first order optimality conditions, and regularity estimates for the optimal variables. We propose two strategies to discretize the fractional optimal control problem: a semidiscrete approach where the control is not discretized - the so-called variational discretization approach - and a fully discrete approach where the control variable is discretized with piecewise constant functions. Both schemes rely on the discreti…
This paper presents a fractional-order framework for control of blood pressure. A new perspective is explored to control the blood pressure in lieu of the conventional control framework. A multi-variable scenario is adopted to control two outputs: Mean Arterial Blood Pressure (MABP) and Cardiac Output (CO) simultaneously.
In this paper we will present a mathematical description and analysis of a fractional-order regulated system in the state space and the state-space controller design based on placing the closed-loop poles on the complex plane. Presented are the results of simulations and stability investigation of this system.
We consider a terminal control problem for processes governed by a nonlinear system of fractional ODEs. In order to show existence of the control, we first consider the linear counterpart of the system and reprove a number of classical theorems in the fractional setting (representation of the solution through the Gramian type matrix, Kalman's principle, equivalence of the controllability and observability). We are then in the position to use a fixed point theorem approach and various techniques from the fractional calculus theory to get the desired result.
We establish existence, approximate controllability and optimal control of a class of impulsive non-local non-linear fractional dynamical systems in Banach spaces. We use fractional calculus, sectorial operators and Krasnoselskii fixed point theorems for the main results. Approximate controllability results are discussed with respect to the inhomogeneous non-linear part. Moreover, we prove existence results of optimal pairs of corresponding fractional control systems with a Bolza cost functional.
We prove existence and uniqueness of mild solutions to Sobolev type fractional nonlocal dynamic equations in Banach spaces. The Sobolev nonlocal condition is considered in terms of a Riemann-Liouville fractional derivative. A Lagrange optimal control problem is considered, and existence of a multi-integral solution obtained. Main tools include fractional calculus, semigroup theory, fractional power of operators, a singular version of Gronwall's inequality, and Leray-Schauder fixed point theorem. An example illustrating the theory is given.
A non-convex control system governed by a nonlinear impulsive evolution equation of Hilfer fractional order in a Banach space is considered. The existence of admissible state-control pair is established. Then the introduction of suitable measure-valued control convexifies the system, and the relaxed system is obtained. Further, the relaxation theorem for the described class is proved along with the existence of optimal relaxed control.
The paper is devoted to the development of control procedures with a guide for conflict-controlled dynamical systems described by ordinary fractional differential equations with the Caputo derivative of an order $α\in (0, 1).$ For the case when the guide is in a certain sense a copy of the system, a mutual aiming procedure between the initial system and the guide is elaborated. The proof of proximity between motions of the systems is based on the estimate of the fractional derivative of the superposition of a convex Lyapunov function and a function represented by the fractional integral of an …
The study of fractional variational problems with derivatives in the sense of Caputo is a recent subject, the main results being Agrawal's necessary optimality conditions of Euler-Lagrange and respective transversality conditions. Using Agrawal's Euler-Lagrange equation and the Lagrange multiplier technique, we obtain here a Noether-like theorem for fractional optimal control problems in the sense of Caputo.
We consider a mean-field optimal control problem for stochastic differential equations with delay driven by fractional Brownian motion with Hurst parameter greater than one half. Stochastic optimal control problems driven by fractional Brownian motion can not be studied using classical methods, because the fractional Brownian motion is neither a Markov process nor a semi-martingale. However, using the fractional White noise calculus combined with some special tools related to the differentiation for functions of measures, we establish and prove necessary and sufficient stochastic maximum princ…
The goal of this work is to prove global controllability and stabilization properties for the fractional Schrödinger equation on $d$-dimensional compact Riemannian manifolds without boundary $(M,g)$. To prove our main results we use techniques of pseudo-differential calculus on manifolds. More precisely, by using microlocal analysis, we are able to prove propagation of regularity which together with the so-called Geometric Control Condition and Unique Continuation Property help us to prove global control results for the system under consideration. As a main novelty this manuscript presents the…
In this paper an offset-free model predictive control scheme is presented for fractional-order systems using the Grünwald-Letnikov derivative. The infinite-history fractional-order system is approximated by a finite-dimensional state-space system and the modeling error is cast as a bounded disturbance term. Using a state observer, it is shown that the unknown disturbance at steady state can be reconstructed and modeling errors and other persistent disturbances can be attenuated. The effectiveness of the proposed controller-observer ensemble is demonstrated in the optimal administration of an a…
In this article, we develop a simple mathematical GNU Octave/MATLAB code that is easy to modify for the simulation of mathematical models governed by fractional-order differential equations, and for the resolution of fractional-order optimal control problems through Pontryagin's maximum principle (indirect approach to optimal control). For this purpose, a fractional-order model for the respiratory syncytial virus (RSV) infection is considered. The model is an improvement of one first proposed by the authors in [Chaos Solitons Fractals 117 (2018), 142--149]. The initial value problem associated…
The article is dedicated towards the study of fractional order non-linear differential systems with non-instantaneous impulses involving Riemann-Liouville derivatives with fixed lower limit and appropriate integral type initial conditions in Banach spaces. First, mild solution of the system is constructed and subsequently its existence is proven using Banach's fixed point theorem. Then, results of approximate controllability are established using concept of fractional semigroup and an iterative technique. Suitable examples are given in the end supporting the methodology along with pointing out…
